The operator tag is used to name a company, corporation, person or any other entity who is directly in charge of the current operation of a map object.

When choosing the appropriate value for the operator=* tag, it is beneficial to use exactly the same text including capitalization across all entities managed by the same structure. Please refer to taginfo, or the title of the associated Wikipedia article for the organization as a starting point.

If the vast majority of a certain object in an area is operated by a certain organization and only very few by others then it may be sufficient to only tag the exceptions.
For example, when nearly all roads in an area are managed by a local authority then it would be sufficient to only tag those that are not with an operator tag.

Contents

Owner or brand name aren't necessarily the operator

An operator isn't necessarily the owner of the map feature. Many chains (store, restaurants...) use a franchise system, where the brand does not operate the point-of-presence.
For example, a lot of transport and communication networks had been built by public administrations and are now operated by private companies.
Be sure to use this tag according to this criteria.

Sometimes, it's more complicated. In this example; the company that manages the hotel is not the same company that owns the building. In other cases; there is another party between the owner and management company that rents or leases the building.

Postal Services

Due to the emerging of private postal service providers, it may be useful for map readers to know which postal services provider operates a certain amenity=post_box.
In Germany there are already some private companies like PIN or Stadtbrief who install their own post boxes. Who wants to ship a letter has to choose the post box of "his" postal service provider ?

Other

Other examples are pubs, restaurants and hotel chains, maybe also streets maintained by private companies where a fee is required.